Warning: Here Be Nerdiness

Yesterday, a conversation with a student somehow turned to the Michael Bay movie Armageddon–a film that shares the same real estate in my cortex as the Tuck Rule Game, the George W. Bush presidency, and Star Wars Episode II. We both agreed that the idea of a single nuclear weapon splitting a mass as gargantuanly hippopotamic as an “asteroid the size of Texas” and accelerating the hemispheres away from the earth failed to pass even a lunatic’s laugh test, but my student seemed convinced that all of humanity’s nuclear arsenal put together could do the job, though it would be unwieldily.

Nerd that I am, I decided to look into the idea. How much energy would be required to split an asteroid 1,300 km in diameter (roughly the width of Texas) and drive the pieces apart at a speed that would allow them to escape the former body’s own gravity? And could such energies be produced by the combined arsenals of the world’s nuclear powers? (For simplicity’s sake, I’m ignoring that the hemispheres would also have to escape the Earth’s gravity, which would be substantial since at the time of the split the asteroid is supposedly four hours away from impacting us. We really shouldn’t ignore this, because it does us little good to split these masses if all it means is that we’ll be eventually hit by two of them–in addition to whatever 1-10km shrapnel pieces that would almost surely fly off–instead of one, so in reality we’d have to accelerate these hemispheres to a velocity that would allow them to escape not only each other, but also Earth. But you’ll see in a moment that for this purpose, we don’t have to be all that exact. Orders of magnitude for comparison will suffice to tell us what we want to know.)

The thing we’re dealing with here is gravitational binding energy. Basically, gravitational attraction binds the asteroid together. Breaking those connections takes energy. Now gravity is a weak force, but in a heavy body, it’s substantial. Now we’re going to be lazy and assume that the asteroid, which I’ll call Bay-2012, has a uniform density. (It almost surely wouldn’t, but, as I said earlier, we don’t need to be all that exact.) We’ll also assume it’s spheroid because any asteroid that size would be thanks to gravity. Let’s say the asteroid is as dense as the largest asteroid we actually know of: Ceres. Ceres clocks in at roughly 2.7 g/cubic cm. We can use that and the asteroid’s radius–roughly 650 km, to work out the asteroid’s mass by taking advantage of the following formula:

Bay-2012’s mass by this formula: 9.4E+20 kg

Now, using the gravitational binding energy formula for a uniform sphere, we can work out what it’ll take to blow Bay-2012 to smithereens. (A very Bayesque result.)

How much energy could we expect from all the nuclear weapons on the planet? (We’ll ignore the problems of moving all these weapons, quickly and simultaneously, into the middle of the asteroid for detonation.) We don’t know exactly how many weapons there are, but most estimates I’ve seen suggest that human beings have amassed some 19,000 nuclear warheads. Now these weapons come in a variety of different yields, but lets assume an average of 1 MT (roughly 4E+15 Joules). 19,000 of them detonated simultaneously within Bay-2012 would produce 7.6E+19 Joules, or roughly 1/263,000th the energy required to separate the two hemispheres from each other. It’s a bit like trying to break your car door open with a toothpick. In reality, it would take 1-3 billion 1 MT bombs to pull off Armageddon‘s stunt. (The last thing we’ll ignore is the effect on the Earth of such a titanic energy release within the boundaries of the moon’s orbit.)

The movies–Bay is hardly the only guilty flickmaker–reflect our own tendency to overestimate what nuclear weapons are capable of. Here’s the thing: nuclear bombs are frighteningly effective at killing us, which is why we should try to avoid deploying them. They could easily end our civilization. But they’re not capable of wiping out large planetary bodies. For that, you need this:

Fortunately, Dick Cheney won’t be finished with that until he’s on at least his third clone body, so we’re safe for now.